View source: R/chaAllPairsNashimotoTest.R
chaAllPairsNashimotoTest | R Documentation |
Performs Nashimoto and Wright's all-pairs comparison procedure for simply ordered mean ranksums (NPT'-test and NPY'-test).
According to the authors, the procedure shall only be
applied after Chacko's test (see chackoTest
) indicates
global significance.
chaAllPairsNashimotoTest(x, ...)
## Default S3 method:
chaAllPairsNashimotoTest(
x,
g,
p.adjust.method = c(p.adjust.methods),
alternative = c("greater", "less"),
dist = c("Normal", "h"),
...
)
## S3 method for class 'formula'
chaAllPairsNashimotoTest(
formula,
data,
subset,
na.action,
p.adjust.method = c(p.adjust.methods),
alternative = c("greater", "less"),
dist = c("Normal", "h"),
...
)
x |
a numeric vector of data values, or a list of numeric data vectors. |
... |
further arguments to be passed to or from methods. |
g |
a vector or factor object giving the group for the
corresponding elements of |
p.adjust.method |
method for adjusting p values. Ignored if |
alternative |
the alternative hypothesis. Defaults to |
dist |
the test distribution. Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
subset |
an optional vector specifying a subset of observations to be used. |
na.action |
a function which indicates what should happen when
the data contain |
The modified procedure uses the property of a simple order,
\theta_m' - \theta_m \le \theta_j - \theta_i \le \theta_l' - \theta_l
\qquad (l \le i \le m~\mathrm{and}~ m' \le j \le l')
.
The null hypothesis H_{ij}: \theta_i = \theta_j
is tested against
the alternative A_{ij}: \theta_i < \theta_j
for any
1 \le i < j \le k
.
Let R_{ij}
be the rank of X_{ij}
,
where X_{ij}
is jointly ranked
from \left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i
,
then the test statistics for all-pairs comparisons
and a balanced design is calculated as
\hat{T}_{ij} = \max_{i \le m < m' \le j}
\frac{\left(\bar{R}_{m'} - \bar{R}_m \right)}
{\sigma_a / \sqrt{n}},
with n = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k)
, \bar{R}_i
the mean rank for the i
th group,
and the expected variance (without ties) \sigma_a^2 = N \left(N + 1 \right) / 12
.
For the NPY'-test (dist = "h"
), if T_{ij} > h_{k-1,\alpha,\infty}
.
For the unbalanced case with moderate imbalance the test statistic is
\hat{T}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)}
{\sigma_a \left(1/n_m + 1/n_{m'}\right)^{1/2}},
For the NPY'-test (dist="h"
) the null hypothesis is rejected in an unbalanced design,
if \hat{T}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}
.
In case of a NPY'-test, the function does not return p-values. Instead the critical h-values
as given in the tables of Hayter (1990) for \alpha = 0.05
(one-sided)
are looked up according to the number of groups (k-1
) and
the degree of freedoms (v = \infty
).
For the NPT'-test (dist = "Normal"
), the null hypothesis is rejected, if
T_{ij} > \sqrt{2} t_{\alpha,\infty} = \sqrt{2} z_\alpha
. Although Nashimoto and Wright (2005) originally did not use any p-adjustment,
any method as available by p.adjust.methods
can
be selected for the adjustment of p-values estimated from
the standard normal distribution.
Either a list of class "osrt"
if dist = "h"
or a list
of class "PMCMR"
if dist = "Normal"
.
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated statistic(s)
critical values for \alpha = 0.05
.
a character string describing the alternative hypothesis.
the parameter(s) of the test distribution.
a string that denotes the test distribution.
There are print and summary methods available.
A list with class "PMCMR"
containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
The function will give a warning for the unbalanced case and returns the
critical value h_{k-1,\alpha,\infty} / \sqrt{2}
if applicable.
Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, J Amer Stat Assoc 85, 778–785.
Nashimoto, K., Wright, F.T. (2007) Nonparametric Multiple-Comparison Methods for Simply Ordered Medians. Comput Stat Data Anal 51, 5068–5076.
Normal
, chackoTest
,
NPMTest
## Example from Shirley (1977)
## Reaction times of mice to stimuli to their tails.
x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3,
2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8,
7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4,
9, 8.4, 2.4, 7.8)
g <- gl(4, 10)
## Shirley's test
## one-sided test using look-up table
shirleyWilliamsTest(x ~ g, alternative = "greater")
## Chacko's global hypothesis test for 'greater'
chackoTest(x , g)
## post-hoc test, default is standard normal distribution (NPT'-test)
summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none"))
## same but h-distribution (NPY'-test)
chaAllPairsNashimotoTest(x, g, dist = "h")
## NPM-test
NPMTest(x, g)
## Hayter-Stone test
hayterStoneTest(x, g)
## all-pairs comparisons
hsAllPairsTest(x, g)
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